This all sounds very similar to the problem I've had with my 1992 Univega mountain bike's triple using Shimano Rapidfire thumb shifters (all thumbies, not thumb/trigger). Upshifts were fine. Downshifts from the middle chain ring to the smallest were fine. But I couldn't get it to shift properly from the big ring to the middle. It would hang or skip over the middle to the smallest chain ring. Very annoying when trying to maintain speed on climbs.
I removed the shifter pod three times for cleaning, without a complete disassembly down to the loose individual parts. No improvment. It seems to function properly until mounted on the bike and under tension.
I finally got tired of messing with it and replaced it with the cheap Sunrace SLM10 friction shifter, just for the left side/front derailer. Works fine, and eliminated the derailer itself as any source of the problem.
Eventually I'll try to find time to completely disassemble the Rapidfire shifter and see if I can get it working. Or maybe I'll try to find a new old stock shifter.
Meanwhile, try a Sunrace SLM10 friction shifter just on the left side for the front derailer. See if that works.
You can buy a complete
Sunrace SLM10 set for $9 with free shipping from Amazon (for Prime subscribers) -- both shifters, cables (cheap galvanized but functional for at least a few months, depending on the weather) and cable housings with end caps. Quick and easy to install. Just dial out any chain rub with the shifter lever. After a test ride you'll figure out the right compromise in the thumbscrew adjuster tension for ease of shifting while holding the derailer without springing back under tension.